\(\int \frac {c+d x^3+e x^6+f x^9}{x^{10} (a+b x^3)^3} \, dx\) [284]

Optimal result

Integrand size = 30, antiderivative size = 218 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{10} \left (a+b x^3\right )^3} \, dx=-\frac {c}{9 a^3 x^9}+\frac {3 b c-a d}{6 a^4 x^6}-\frac {6 b^2 c-3 a b d+a^2 e}{3 a^5 x^3}-\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{6 a^4 \left (a+b x^3\right )^2}-\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{3 a^5 \left (a+b x^3\right )}-\frac {\left (10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f\right ) \log (x)}{a^6}+\frac {\left (10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f\right ) \log \left (a+b x^3\right )}{3 a^6} \]

[Out]

-1/9*c/a^3/x^9+1/6*(-a*d+3*b*c)/a^4/x^6+1/3*(-a^2*e+3*a*b*d-6*b^2*c)/a^5/x^3+1/6*(a^3*f-a^2*b*e+a*b^2*d-b^3*c)
/a^4/(b*x^3+a)^2+1/3*(a^3*f-2*a^2*b*e+3*a*b^2*d-4*b^3*c)/a^5/(b*x^3+a)-(-a^3*f+3*a^2*b*e-6*a*b^2*d+10*b^3*c)*l
n(x)/a^6+1/3*(-a^3*f+3*a^2*b*e-6*a*b^2*d+10*b^3*c)*ln(b*x^3+a)/a^6

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1835, 1634} \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{10} \left (a+b x^3\right )^3} \, dx=\frac {3 b c-a d}{6 a^4 x^6}-\frac {c}{9 a^3 x^9}-\frac {a^2 e-3 a b d+6 b^2 c}{3 a^5 x^3}+\frac {\log \left (a+b x^3\right ) \left (a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c\right )}{3 a^6}-\frac {\log (x) \left (a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c\right )}{a^6}-\frac {a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{3 a^5 \left (a+b x^3\right )}-\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{6 a^4 \left (a+b x^3\right )^2} \]

[In]

Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^10*(a + b*x^3)^3),x]

[Out]

-1/9*c/(a^3*x^9) + (3*b*c - a*d)/(6*a^4*x^6) - (6*b^2*c - 3*a*b*d + a^2*e)/(3*a^5*x^3) - (b^3*c - a*b^2*d + a^
2*b*e - a^3*f)/(6*a^4*(a + b*x^3)^2) - (4*b^3*c - 3*a*b^2*d + 2*a^2*b*e - a^3*f)/(3*a^5*(a + b*x^3)) - ((10*b^
3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)*Log[x])/a^6 + ((10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)*Log[a + b*x^3])
/(3*a^6)

Rule 1634

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rule 1835

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] -
1)*SubstFor[x^n, Pq, x]*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && PolyQ[Pq, x^n] && Intege
rQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {c+d x+e x^2+f x^3}{x^4 (a+b x)^3} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (\frac {c}{a^3 x^4}+\frac {-3 b c+a d}{a^4 x^3}+\frac {6 b^2 c-3 a b d+a^2 e}{a^5 x^2}+\frac {-10 b^3 c+6 a b^2 d-3 a^2 b e+a^3 f}{a^6 x}-\frac {b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^4 (a+b x)^3}-\frac {b \left (-4 b^3 c+3 a b^2 d-2 a^2 b e+a^3 f\right )}{a^5 (a+b x)^2}-\frac {b \left (-10 b^3 c+6 a b^2 d-3 a^2 b e+a^3 f\right )}{a^6 (a+b x)}\right ) \, dx,x,x^3\right ) \\ & = -\frac {c}{9 a^3 x^9}+\frac {3 b c-a d}{6 a^4 x^6}-\frac {6 b^2 c-3 a b d+a^2 e}{3 a^5 x^3}-\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{6 a^4 \left (a+b x^3\right )^2}-\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{3 a^5 \left (a+b x^3\right )}-\frac {\left (10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f\right ) \log (x)}{a^6}+\frac {\left (10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f\right ) \log \left (a+b x^3\right )}{3 a^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.92 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{10} \left (a+b x^3\right )^3} \, dx=\frac {-\frac {2 a^3 c}{x^9}-\frac {3 a^2 (-3 b c+a d)}{x^6}-\frac {6 a \left (6 b^2 c-3 a b d+a^2 e\right )}{x^3}+\frac {3 a^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{\left (a+b x^3\right )^2}+\frac {6 a \left (-4 b^3 c+3 a b^2 d-2 a^2 b e+a^3 f\right )}{a+b x^3}+18 \left (-10 b^3 c+6 a b^2 d-3 a^2 b e+a^3 f\right ) \log (x)+6 \left (10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f\right ) \log \left (a+b x^3\right )}{18 a^6} \]

[In]

Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^10*(a + b*x^3)^3),x]

[Out]

((-2*a^3*c)/x^9 - (3*a^2*(-3*b*c + a*d))/x^6 - (6*a*(6*b^2*c - 3*a*b*d + a^2*e))/x^3 + (3*a^2*(-(b^3*c) + a*b^
2*d - a^2*b*e + a^3*f))/(a + b*x^3)^2 + (6*a*(-4*b^3*c + 3*a*b^2*d - 2*a^2*b*e + a^3*f))/(a + b*x^3) + 18*(-10
*b^3*c + 6*a*b^2*d - 3*a^2*b*e + a^3*f)*Log[x] + 6*(10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)*Log[a + b*x^3])/
(18*a^6)

Maple [A] (verified)

Time = 1.53 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.97

method	result	size
default	\(-\frac {c}{9 a^{3} x^{9}}-\frac {a d -3 b c}{6 a^{4} x^{6}}-\frac {a^{2} e -3 a b d +6 b^{2} c}{3 a^{5} x^{3}}+\frac {\left (f \,a^{3}-3 a^{2} b e +6 a \,b^{2} d -10 b^{3} c \right ) \ln \left (x \right )}{a^{6}}-\frac {b \left (\frac {\left (f \,a^{3}-3 a^{2} b e +6 a \,b^{2} d -10 b^{3} c \right ) \ln \left (b \,x^{3}+a \right )}{b}-\frac {a^{2} \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right )}{2 b \left (b \,x^{3}+a \right )^{2}}-\frac {a \left (f \,a^{3}-2 a^{2} b e +3 a \,b^{2} d -4 b^{3} c \right )}{b \left (b \,x^{3}+a \right )}\right )}{3 a^{6}}\)	\(212\)
norman	\(\frac {-\frac {c}{9 a}-\frac {\left (3 a d -5 b c \right ) x^{3}}{18 a^{2}}-\frac {\left (3 a^{2} e -6 a b d +10 b^{2} c \right ) x^{6}}{9 a^{3}}+\frac {\left (a^{3} b^{2} f -3 a^{2} b^{3} e +6 a \,b^{4} d -10 b^{5} c \right ) x^{9}}{2 a^{4} b^{2}}+\frac {\left (a^{3} b^{2} f -3 a^{2} b^{3} e +6 a \,b^{4} d -10 b^{5} c \right ) x^{12}}{3 a^{5} b}}{x^{9} \left (b \,x^{3}+a \right )^{2}}+\frac {\left (f \,a^{3}-3 a^{2} b e +6 a \,b^{2} d -10 b^{3} c \right ) \ln \left (x \right )}{a^{6}}-\frac {\left (f \,a^{3}-3 a^{2} b e +6 a \,b^{2} d -10 b^{3} c \right ) \ln \left (b \,x^{3}+a \right )}{3 a^{6}}\)	\(220\)
risch	\(\frac {-\frac {c}{9 a}-\frac {\left (3 a d -5 b c \right ) x^{3}}{18 a^{2}}-\frac {\left (3 a^{2} e -6 a b d +10 b^{2} c \right ) x^{6}}{9 a^{3}}+\frac {\left (f \,a^{3}-3 a^{2} b e +6 a \,b^{2} d -10 b^{3} c \right ) x^{9}}{2 a^{4}}+\frac {b \left (f \,a^{3}-3 a^{2} b e +6 a \,b^{2} d -10 b^{3} c \right ) x^{12}}{3 a^{5}}}{x^{9} \left (b \,x^{3}+a \right )^{2}}+\frac {\ln \left (x \right ) f}{a^{3}}-\frac {3 \ln \left (x \right ) b e}{a^{4}}+\frac {6 \ln \left (x \right ) b^{2} d}{a^{5}}-\frac {10 \ln \left (x \right ) b^{3} c}{a^{6}}-\frac {\ln \left (b \,x^{3}+a \right ) f}{3 a^{3}}+\frac {\ln \left (b \,x^{3}+a \right ) b e}{a^{4}}-\frac {2 \ln \left (b \,x^{3}+a \right ) b^{2} d}{a^{5}}+\frac {10 \ln \left (b \,x^{3}+a \right ) b^{3} c}{3 a^{6}}\)	\(234\)
parallelrisch	\(\frac {-6 x^{6} a^{5} b^{2} e +12 x^{6} a^{4} b^{3} d -20 x^{6} a^{3} b^{4} c -90 x^{9} a^{2} b^{5} c +9 x^{9} a^{5} b^{2} f -27 x^{9} a^{4} b^{3} e -3 x^{3} a^{5} b^{2} d +5 x^{3} a^{4} b^{3} c +54 x^{9} a^{3} b^{4} d -12 \ln \left (b \,x^{3}+a \right ) x^{12} a^{4} b^{3} f -2 c \,b^{2} a^{5}+36 \ln \left (b \,x^{3}+a \right ) x^{12} a^{3} b^{4} e +18 \ln \left (b \,x^{3}+a \right ) x^{15} a^{2} b^{5} e -36 \ln \left (b \,x^{3}+a \right ) x^{15} a \,b^{6} d +36 \ln \left (x \right ) x^{12} a^{4} b^{3} f -108 \ln \left (x \right ) x^{12} a^{3} b^{4} e +216 \ln \left (x \right ) x^{12} a^{2} b^{5} d -360 \ln \left (x \right ) x^{12} a \,b^{6} c -72 \ln \left (b \,x^{3}+a \right ) x^{12} a^{2} b^{5} d +120 \ln \left (b \,x^{3}+a \right ) x^{12} a \,b^{6} c -180 \ln \left (x \right ) x^{15} b^{7} c +18 \ln \left (x \right ) x^{15} a^{3} b^{4} f -54 \ln \left (x \right ) x^{15} a^{2} b^{5} e +108 \ln \left (x \right ) x^{15} a \,b^{6} d -6 \ln \left (b \,x^{3}+a \right ) x^{15} a^{3} b^{4} f +18 \ln \left (x \right ) x^{9} a^{5} b^{2} f +60 \ln \left (b \,x^{3}+a \right ) x^{15} b^{7} c +36 x^{12} a^{2} b^{5} d +6 x^{12} a^{4} b^{3} f -18 x^{12} a^{3} b^{4} e -54 \ln \left (x \right ) x^{9} a^{4} b^{3} e -60 x^{12} a \,b^{6} c +108 \ln \left (x \right ) x^{9} a^{3} b^{4} d -180 \ln \left (x \right ) x^{9} a^{2} b^{5} c -6 \ln \left (b \,x^{3}+a \right ) x^{9} a^{5} b^{2} f +18 \ln \left (b \,x^{3}+a \right ) x^{9} a^{4} b^{3} e -36 \ln \left (b \,x^{3}+a \right ) x^{9} a^{3} b^{4} d +60 \ln \left (b \,x^{3}+a \right ) x^{9} a^{2} b^{5} c}{18 a^{6} b^{2} x^{9} \left (b \,x^{3}+a \right )^{2}}\)	\(579\)

[In]

int((f*x^9+e*x^6+d*x^3+c)/x^10/(b*x^3+a)^3,x,method=_RETURNVERBOSE)

[Out]

-1/9*c/a^3/x^9-1/6*(a*d-3*b*c)/a^4/x^6-1/3*(a^2*e-3*a*b*d+6*b^2*c)/a^5/x^3+(a^3*f-3*a^2*b*e+6*a*b^2*d-10*b^3*c
)/a^6*ln(x)-1/3*b/a^6*((a^3*f-3*a^2*b*e+6*a*b^2*d-10*b^3*c)/b*ln(b*x^3+a)-1/2*a^2*(a^3*f-a^2*b*e+a*b^2*d-b^3*c
)/b/(b*x^3+a)^2-a*(a^3*f-2*a^2*b*e+3*a*b^2*d-4*b^3*c)/b/(b*x^3+a))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.82 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{10} \left (a+b x^3\right )^3} \, dx=-\frac {6 \, {\left (10 \, a b^{4} c - 6 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e - a^{4} b f\right )} x^{12} + 9 \, {\left (10 \, a^{2} b^{3} c - 6 \, a^{3} b^{2} d + 3 \, a^{4} b e - a^{5} f\right )} x^{9} + 2 \, {\left (10 \, a^{3} b^{2} c - 6 \, a^{4} b d + 3 \, a^{5} e\right )} x^{6} + 2 \, a^{5} c - {\left (5 \, a^{4} b c - 3 \, a^{5} d\right )} x^{3} - 6 \, {\left ({\left (10 \, b^{5} c - 6 \, a b^{4} d + 3 \, a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{15} + 2 \, {\left (10 \, a b^{4} c - 6 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e - a^{4} b f\right )} x^{12} + {\left (10 \, a^{2} b^{3} c - 6 \, a^{3} b^{2} d + 3 \, a^{4} b e - a^{5} f\right )} x^{9}\right )} \log \left (b x^{3} + a\right ) + 18 \, {\left ({\left (10 \, b^{5} c - 6 \, a b^{4} d + 3 \, a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{15} + 2 \, {\left (10 \, a b^{4} c - 6 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e - a^{4} b f\right )} x^{12} + {\left (10 \, a^{2} b^{3} c - 6 \, a^{3} b^{2} d + 3 \, a^{4} b e - a^{5} f\right )} x^{9}\right )} \log \left (x\right )}{18 \, {\left (a^{6} b^{2} x^{15} + 2 \, a^{7} b x^{12} + a^{8} x^{9}\right )}} \]

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^10/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

-1/18*(6*(10*a*b^4*c - 6*a^2*b^3*d + 3*a^3*b^2*e - a^4*b*f)*x^12 + 9*(10*a^2*b^3*c - 6*a^3*b^2*d + 3*a^4*b*e -
a^5*f)*x^9 + 2*(10*a^3*b^2*c - 6*a^4*b*d + 3*a^5*e)*x^6 + 2*a^5*c - (5*a^4*b*c - 3*a^5*d)*x^3 - 6*((10*b^5*c
- 6*a*b^4*d + 3*a^2*b^3*e - a^3*b^2*f)*x^15 + 2*(10*a*b^4*c - 6*a^2*b^3*d + 3*a^3*b^2*e - a^4*b*f)*x^12 + (10*
a^2*b^3*c - 6*a^3*b^2*d + 3*a^4*b*e - a^5*f)*x^9)*log(b*x^3 + a) + 18*((10*b^5*c - 6*a*b^4*d + 3*a^2*b^3*e - a
^3*b^2*f)*x^15 + 2*(10*a*b^4*c - 6*a^2*b^3*d + 3*a^3*b^2*e - a^4*b*f)*x^12 + (10*a^2*b^3*c - 6*a^3*b^2*d + 3*a
^4*b*e - a^5*f)*x^9)*log(x))/(a^6*b^2*x^15 + 2*a^7*b*x^12 + a^8*x^9)

Sympy [F(-1)]

Timed out. \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{10} \left (a+b x^3\right )^3} \, dx=\text {Timed out} \]

[In]

integrate((f*x**9+e*x**6+d*x**3+c)/x**10/(b*x**3+a)**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.06 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{10} \left (a+b x^3\right )^3} \, dx=-\frac {6 \, {\left (10 \, b^{4} c - 6 \, a b^{3} d + 3 \, a^{2} b^{2} e - a^{3} b f\right )} x^{12} + 9 \, {\left (10 \, a b^{3} c - 6 \, a^{2} b^{2} d + 3 \, a^{3} b e - a^{4} f\right )} x^{9} + 2 \, {\left (10 \, a^{2} b^{2} c - 6 \, a^{3} b d + 3 \, a^{4} e\right )} x^{6} + 2 \, a^{4} c - {\left (5 \, a^{3} b c - 3 \, a^{4} d\right )} x^{3}}{18 \, {\left (a^{5} b^{2} x^{15} + 2 \, a^{6} b x^{12} + a^{7} x^{9}\right )}} + \frac {{\left (10 \, b^{3} c - 6 \, a b^{2} d + 3 \, a^{2} b e - a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{6}} - \frac {{\left (10 \, b^{3} c - 6 \, a b^{2} d + 3 \, a^{2} b e - a^{3} f\right )} \log \left (x^{3}\right )}{3 \, a^{6}} \]

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^10/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

-1/18*(6*(10*b^4*c - 6*a*b^3*d + 3*a^2*b^2*e - a^3*b*f)*x^12 + 9*(10*a*b^3*c - 6*a^2*b^2*d + 3*a^3*b*e - a^4*f
)*x^9 + 2*(10*a^2*b^2*c - 6*a^3*b*d + 3*a^4*e)*x^6 + 2*a^4*c - (5*a^3*b*c - 3*a^4*d)*x^3)/(a^5*b^2*x^15 + 2*a^
6*b*x^12 + a^7*x^9) + 1/3*(10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)*log(b*x^3 + a)/a^6 - 1/3*(10*b^3*c - 6*a*
b^2*d + 3*a^2*b*e - a^3*f)*log(x^3)/a^6

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.45 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{10} \left (a+b x^3\right )^3} \, dx=-\frac {{\left (10 \, b^{3} c - 6 \, a b^{2} d + 3 \, a^{2} b e - a^{3} f\right )} \log \left ({\left | x \right |}\right )}{a^{6}} + \frac {{\left (10 \, b^{4} c - 6 \, a b^{3} d + 3 \, a^{2} b^{2} e - a^{3} b f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{6} b} - \frac {30 \, b^{5} c x^{6} - 18 \, a b^{4} d x^{6} + 9 \, a^{2} b^{3} e x^{6} - 3 \, a^{3} b^{2} f x^{6} + 68 \, a b^{4} c x^{3} - 42 \, a^{2} b^{3} d x^{3} + 22 \, a^{3} b^{2} e x^{3} - 8 \, a^{4} b f x^{3} + 39 \, a^{2} b^{3} c - 25 \, a^{3} b^{2} d + 14 \, a^{4} b e - 6 \, a^{5} f}{6 \, {\left (b x^{3} + a\right )}^{2} a^{6}} + \frac {110 \, b^{3} c x^{9} - 66 \, a b^{2} d x^{9} + 33 \, a^{2} b e x^{9} - 11 \, a^{3} f x^{9} - 36 \, a b^{2} c x^{6} + 18 \, a^{2} b d x^{6} - 6 \, a^{3} e x^{6} + 9 \, a^{2} b c x^{3} - 3 \, a^{3} d x^{3} - 2 \, a^{3} c}{18 \, a^{6} x^{9}} \]

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^10/(b*x^3+a)^3,x, algorithm="giac")

[Out]

-(10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)*log(abs(x))/a^6 + 1/3*(10*b^4*c - 6*a*b^3*d + 3*a^2*b^2*e - a^3*b*
f)*log(abs(b*x^3 + a))/(a^6*b) - 1/6*(30*b^5*c*x^6 - 18*a*b^4*d*x^6 + 9*a^2*b^3*e*x^6 - 3*a^3*b^2*f*x^6 + 68*a
*b^4*c*x^3 - 42*a^2*b^3*d*x^3 + 22*a^3*b^2*e*x^3 - 8*a^4*b*f*x^3 + 39*a^2*b^3*c - 25*a^3*b^2*d + 14*a^4*b*e -
6*a^5*f)/((b*x^3 + a)^2*a^6) + 1/18*(110*b^3*c*x^9 - 66*a*b^2*d*x^9 + 33*a^2*b*e*x^9 - 11*a^3*f*x^9 - 36*a*b^2
*c*x^6 + 18*a^2*b*d*x^6 - 6*a^3*e*x^6 + 9*a^2*b*c*x^3 - 3*a^3*d*x^3 - 2*a^3*c)/(a^6*x^9)

Mupad [B] (verification not implemented)

Time = 9.50 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.02 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{10} \left (a+b x^3\right )^3} \, dx=\frac {\ln \left (b\,x^3+a\right )\,\left (-f\,a^3+3\,e\,a^2\,b-6\,d\,a\,b^2+10\,c\,b^3\right )}{3\,a^6}-\frac {\frac {c}{9\,a}+\frac {x^9\,\left (-f\,a^3+3\,e\,a^2\,b-6\,d\,a\,b^2+10\,c\,b^3\right )}{2\,a^4}+\frac {x^3\,\left (3\,a\,d-5\,b\,c\right )}{18\,a^2}+\frac {x^6\,\left (3\,e\,a^2-6\,d\,a\,b+10\,c\,b^2\right )}{9\,a^3}+\frac {b\,x^{12}\,\left (-f\,a^3+3\,e\,a^2\,b-6\,d\,a\,b^2+10\,c\,b^3\right )}{3\,a^5}}{a^2\,x^9+2\,a\,b\,x^{12}+b^2\,x^{15}}-\frac {\ln \left (x\right )\,\left (-f\,a^3+3\,e\,a^2\,b-6\,d\,a\,b^2+10\,c\,b^3\right )}{a^6} \]

[In]

int((c + d*x^3 + e*x^6 + f*x^9)/(x^10*(a + b*x^3)^3),x)

[Out]

(log(a + b*x^3)*(10*b^3*c - a^3*f - 6*a*b^2*d + 3*a^2*b*e))/(3*a^6) - (c/(9*a) + (x^9*(10*b^3*c - a^3*f - 6*a*
b^2*d + 3*a^2*b*e))/(2*a^4) + (x^3*(3*a*d - 5*b*c))/(18*a^2) + (x^6*(10*b^2*c + 3*a^2*e - 6*a*b*d))/(9*a^3) +
(b*x^12*(10*b^3*c - a^3*f - 6*a*b^2*d + 3*a^2*b*e))/(3*a^5))/(a^2*x^9 + b^2*x^15 + 2*a*b*x^12) - (log(x)*(10*b
^3*c - a^3*f - 6*a*b^2*d + 3*a^2*b*e))/a^6